When transmitting a collection of symbols across a noisy channel, there exists an inherent tradeoff between the power used to transmit each symbol and the length of time or the bandwidth required to transmit the symbol. This tradeoff is shown graphically in the communication efficiency plane of FIG. 1, in which the horizontal axis is the ratio of energy-per-bit to noise power spectral density, E.sub.b /N.sub.o, and the vertical axis is the ratio of the communication rate to bandwidth (bits/sec/Hz). The Shannon limit represents the theoretical maximum capacity of the channel to transfer data as E.sub.b /N.sub.o varies. It is apparent from inspection of FIG. 1 that as more power is used to transmit information, information can be communicated across a channel more rapidly.
The somewhat abstract tradeoff illustrated in FIG. 1 can immediately be understood by considering the case of a speaker attempting to communicate information, for example a telephone number, to a listener across a noisy room. One approach in such a case is for the speaker to shout, thereby rendering the numbers clearly audible to the listener over the noise in the room. Using this method, the speaker can communicate the entire telephone number in a very short time.
Another, far more subtle, approach is for the speaker to speak in a whisper but to elongate the delivery of each syllable. When the speaker transmits the message using this method, the listener can generally pick out the underlying drone of the speaker's message from the background noise, in effect, filtering out the speaker's message Integrating the received signal and noise over a sufficiently long integration interval. The disadvantage of this second approach is that it requires considerably more time to transmit the entire telephone number across the room.
In the context of FIG. 1, the speaker who shouts the phone number across the room operates toward the right-hand side of the communication efficiency plane. As a result, that speaker will be able to operate at a high baud rate and to therefore transmit the entire number very quickly. In contrast, the speaker who whispers the phone number operates toward the left-hand side of the communication efficiency plane and is therefore constrained by the Shannon limit to operate at a low baud rate and to therefore transmit the entire number slowly.
Every communication system can be characterized by an operating point in the communication efficiency plane of FIG. 1. One goal of communication systems design is to place that operating point as close as possible to the Shannon limit. Where bandwidth is a limited resource, one can approach the Shannon limit using a high power system. Such a system does not require large bandwidth for near-error free transmission. Conversely, where power is a limited resource, one can approach the Shannon limit by using a low power system operating over a large bandwidth.
In a wireless communication system, the availability of power directed toward the receiver is limited by the transmitter's power rating and antenna gain. The availability of bandwidth is constrained by the desirability of sharing the available spectrum with as many channels as possible and by FCC governmental and international regulations. Consequently, in a wireless communication system, both power and bandwidth are limited resources.
In addition, a communication system in which different symbols are modulated onto the carrier as different power levels will have certain symbols represented by the lowest power level. Because these symbols are transmitted at lower than average power levels, they will inevitably be more prone to corruption by noise than symbols represented by higher than average power levels. As a result, the error rate associated with the transmission of a message will depend on the content of that message. A higher average power will be required for these symbols to reduce the error rate to a desired level.
Constant power signals are preferred for many wireless systems such as satellite communication systems which typically use more efficient class C operating amplifiers. Because these are non-linear amplifiers that operate at or beyond saturation, a waveform having other than constant amplitude can experience profound distortion as it passes through such an amplifier.
Constant power transmission can be achieved by modulating either the frequency or the phase of a carrier wave. Of these two modulation alternatives, phase modulation is far preferable for satellite communication systems because of its greater bandwidth efficiency. The reason for this greater bandwidth efficiency can be readily understood by considering the operation of a frequency modulation system.
In a frequency modulation system such as frequency shift keying (FSK), each symbol corresponds to a particular frequency. When two symbols correspond to two frequencies that are very close together, the probability of a demodulation error due to noise in the communication channel is high. In order to reduce the probability of such an error, the difference of the two frequencies must exceed a certain fixed amount that is proportional to the channel bandwidth. An increase in the signaling rate entails an increase in the number of frequencies employed. This, of course, consumes bandwidth.
In a conventional phase modulation system, each symbol corresponds to a particular phase angle associated with a carrier having a single frequency. As a result, a large number of symbols can be transmitted without requiring multiple frequencies. Phase modulation systems are thus particularly desirable in applications such as satellite communication systems in which bandwidth is at a premium.
In its simplest form, a phase modulation system operates by transmitting a carrier having a fixed phase angle which is representative of a first symbol during a first time interval. In the context of this application, the "phase angle" of a carrier refers to the principal branch of the arc tangent of the ratio of the imaginary part of the carrier to the real part of the carrier. During a second time interval, the system transmits the carrier but with a fixed phase angle corresponding to a second symbol (not necessarily different from the first symbol). The system then operates at the second phase angle for the duration of the second time interval. The system continues to operate in the foregoing manner until all the symbols that make up the message have been sent. The phase considered as a function of time (hereinafter referred to as the "phaseform") for this system thus traces a discontinuous path in time as shown in the phase cylinder in FIG. 2.
A disadvantage of the foregoing method of operating a phase modulation system is that, it is not possible to efficiently shape the phaseform so that spectral energy will remain concentrated within the allocated bandwidth while maintaining signaling rate. For systems that employ non-overlapping phase symbols, the symbols arc typically phases that are held constant during each symbol transmission interval. For instance, PSK employs constant phase symbols.
Unfortunately, phaseform symbols that have constant values introduce discontinuities whenever a symbol is followed by a different one, and the discontinuity causes energy to spill into adjacent channels, causing adjacent channel interference If the duration of each of the successive non-overlapping symbols is increased to allow shaping for improved energy concentration and decreased adjacent channel interference, then the signaling rate is decreased.
Another approach to overcoming the foregoing disadvantage is to increase the time interval required to transmit a particular symbol but to also allow portions of two or more symbols to be transmitted during each time interval. This type of phase modulation is best understood with reference to FIG. 3, which shows the transmission of six symbols, each represented by a particular phase angle on the vertical axis, and each of which requires three time intervals, .DELTA.T, for transmission.
As shown in FIG. 3, transmission of the first symbol begins at t=0. At the end of the first clock period, t=.DELTA.T (.DELTA.T=clock period I) before the first symbol can be completely transmitted, transmission of the second symbol begins. At the end of the second clock period t=2.DELTA.T, before either the first or second symbol can be completely transmitted, transmission of the third symbol begins. Thus, during the third clock period (between t=2.DELTA.T and t=3.DELTA.T), three symbols are concurrently being transmitted. Since no more than three symbols are transmitted concurrently, a phase modulation system operating as shown in FIG. 3 is said to have a "modulation latency" of three.
It is apparent from FIG. 3 that a phase modulation system having a high latency can, after an initial latency interval of duration equal to the duration of the first symbol, transmit a set of symbols at the same rate as a phase modulation system having a latency of one, notwithstanding the fact that each individual symbol may take longer to transmit. What is less apparent is that the transmission of overlapping symbols in the manner shown in FIG. 3 can be made to eliminate the discontinuities shown in the phase cylinder of FIG. 2 by selecting the phaseform symbols to vary continuously with time. If the individual symbols are continuous, and if the boundaries between adjacent symbols are also continuous, then every linear combination of such symbols overlapped arbitrarily in time will also be continuous. Since it is the discontinuities in the phaseform that result in the spray of high-frequency energy into adjacent channels, phase modulation as shown in FIG. 3 (often referred to as "continuous phase modulation") tends to better confine the signal energy into its allocated bandwidth.
Because energy from two or more overlapping phaseform symbols is commingled, known demodulators cannot readily separate the energy associated with any particular phaseform symbol. In these types of systems, referred to as "partial response" systems, there is less energy per symbol available to the demodulator. In order to compensate for this reduced symbol energy, sequence demodulators, such as Viterbi decoders, are used to recover the symbols. Unfortunately, the use of such sequence demodulators imposes computational costs that increase exponentially with the number of symbols concurrently processed by the demodulator to retrieve the current symbol.
It is known that, in theory, one can separate the individual overlapping symbols that constitute a phaseform if the phaseform is a linear combination of time-shifted since functions (sinc (x)=sin (x)/x ) if the coefficients of the shifted sine functions are known functions of the information symbols. Because the temporally shifted sine functions are orthogonal to one another, it is theoretically possible to recover all the energy associated with a particular symbol, and hence achieve "full response" demodulation, by demodulating the phaseform using a matched filter corresponding to the sine function associated with that symbol. However, this procedure is physically unrealizable because the sine function has infinite duration.
It is therefore desirable to provide systems and methods for achieving physically-realizable full-response continuous phase modulation.